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過去の記録 ～04/19｜次回の予定｜今後の予定 04/20～
| 開催情報 | 木曜日 16:00～17:30 数理科学研究科棟(駒場) 002号室 |
|---|---|
| 担当者 | 石毛 和弘 |
過去の記録
2014年01月23日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Thomas Giletti 氏 (Univ. of Lorraine at Nancy)
Inside dynamics of pushed and pulled fronts (ENGLISH)
Thomas Giletti 氏 (Univ. of Lorraine at Nancy)
Inside dynamics of pushed and pulled fronts (ENGLISH)
[ 講演概要 ]
Mathematical analysis of reaction-diffusion equations is a powerful tool in the understanding of dynamics of many real-life propagation phenomena. A feature of particular interest is the fact that dynamics and their underlying mechanisms vary greatly, depending on the choice of the nonlinearity in the reaction term. In this talk, we will discuss the pushed/pulled front terminology, based upon the role of each component of the front inside the whole propagating structure.
Mathematical analysis of reaction-diffusion equations is a powerful tool in the understanding of dynamics of many real-life propagation phenomena. A feature of particular interest is the fact that dynamics and their underlying mechanisms vary greatly, depending on the choice of the nonlinearity in the reaction term. In this talk, we will discuss the pushed/pulled front terminology, based upon the role of each component of the front inside the whole propagating structure.
2013年12月12日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
安田 勇輝 氏 (東京大学大学院理学系研究科（地球惑星科学専攻）)
くりこみ群の方法による大気重力波の自発的放射メカニズムの解明 (JAPANESE)
安田 勇輝 氏 (東京大学大学院理学系研究科（地球惑星科学専攻）)
くりこみ群の方法による大気重力波の自発的放射メカニズムの解明 (JAPANESE)
[ 講演概要 ]
大気の運動は速いモード (重力波) と遅いモード (地衡流運動) に分けることができる。元の支配方程式系から、遅いモードの相互作用のみを取り出した方程式系をバランスモデルとよぶ。バランスモデルは、重力波を一切含まず、位相空間内の遅い多様体上の運動を記述する。バランスモデルによって大気の大規模運動は良く記述できる。しかし、近年、初期状態が遅い多様体上にあるにも関わらず、後の時間発展と共に重力波が放射され、解の軌道が遅い多様体上から離れることが分かってきた。この現象を重力波の自発的放射とよぶ。
本研究は逓減摂動法の一種であるくりこみ群の方法を用いて、自発的放射を記述する方程式系を導出した。遅いモードと重力波 (速いモード) が効率的に相互作用するためには、時間スケールの一致が必要である。そこで、ドップラー効果を取り込むことで、速いモードが遅い時間スケールを持つことを可能にした。一方で、ドップラー効果とは別に、遅いモード同士の相互作用により、遅い時間スケールを持つ速いモードも励起される。これら二種類の速いモードを別に考えるため、合計三つのモードを導入した。すなわち、遅いモード、「ドップラー効果」により遅い時間スケールを持つ速いモード、「非線形効果」により遅い時間スケールを持つ速いモードである。その上で、くりこみ群の方法を適用し、系の時間発展を記述するくりこみ群方程式系を導出した。くりこみ群方程式系は、遅いモードに従属した成分との準共鳴により、重力波が自発的に放射されることを明らかにする。
気象庁非静力学モデルによる元の支配方程式系の数値積分により、導出したくりこみ群方程式系の妥当性を確認した。さらに、くりこみ群方程式系を用いて、自発的放射の物理的解釈を行った。重力波の放射メカニズムは、山岳波的メカニズムと速度変化メカニズムの二つに分けられ、どちらが主要になるかは、大規模な流れ場の形状によって決定される。また、数値モデルのデータを解析したところ、重力波の振幅は系の無次元パラメータの約 3 乗に比例することがわかった。この結果は、理論的な見積りと整合的であり、実際の解の軌道が遅い多様体からどの程度離れるかの指標を与える。
大気の運動は速いモード (重力波) と遅いモード (地衡流運動) に分けることができる。元の支配方程式系から、遅いモードの相互作用のみを取り出した方程式系をバランスモデルとよぶ。バランスモデルは、重力波を一切含まず、位相空間内の遅い多様体上の運動を記述する。バランスモデルによって大気の大規模運動は良く記述できる。しかし、近年、初期状態が遅い多様体上にあるにも関わらず、後の時間発展と共に重力波が放射され、解の軌道が遅い多様体上から離れることが分かってきた。この現象を重力波の自発的放射とよぶ。
本研究は逓減摂動法の一種であるくりこみ群の方法を用いて、自発的放射を記述する方程式系を導出した。遅いモードと重力波 (速いモード) が効率的に相互作用するためには、時間スケールの一致が必要である。そこで、ドップラー効果を取り込むことで、速いモードが遅い時間スケールを持つことを可能にした。一方で、ドップラー効果とは別に、遅いモード同士の相互作用により、遅い時間スケールを持つ速いモードも励起される。これら二種類の速いモードを別に考えるため、合計三つのモードを導入した。すなわち、遅いモード、「ドップラー効果」により遅い時間スケールを持つ速いモード、「非線形効果」により遅い時間スケールを持つ速いモードである。その上で、くりこみ群の方法を適用し、系の時間発展を記述するくりこみ群方程式系を導出した。くりこみ群方程式系は、遅いモードに従属した成分との準共鳴により、重力波が自発的に放射されることを明らかにする。
気象庁非静力学モデルによる元の支配方程式系の数値積分により、導出したくりこみ群方程式系の妥当性を確認した。さらに、くりこみ群方程式系を用いて、自発的放射の物理的解釈を行った。重力波の放射メカニズムは、山岳波的メカニズムと速度変化メカニズムの二つに分けられ、どちらが主要になるかは、大規模な流れ場の形状によって決定される。また、数値モデルのデータを解析したところ、重力波の振幅は系の無次元パラメータの約 3 乗に比例することがわかった。この結果は、理論的な見積りと整合的であり、実際の解の軌道が遅い多様体からどの程度離れるかの指標を与える。
2013年11月14日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Danielle Hilhorst 氏 (Université de Paris-Sud / CNRS)
Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)
Danielle Hilhorst 氏 (Université de Paris-Sud / CNRS)
Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)
[ 講演概要 ]
We study the singular limit of a damped wave equation with
a bistable nonlinearity. In order to understand interfacial
phenomena, we derive estimates for the generation and the motion
of interfaces. We prove that steep interfaces are generated in
a short time and that their motion is governed by mean curvature
flow under the assumption that the damping is sufficiently strong.
To this purpose, we prove a comparison principle for the damped
wave equation and construct suitable sub- and super-solutions.
This is joint work with Mitsunori Nata.
We study the singular limit of a damped wave equation with
a bistable nonlinearity. In order to understand interfacial
phenomena, we derive estimates for the generation and the motion
of interfaces. We prove that steep interfaces are generated in
a short time and that their motion is governed by mean curvature
flow under the assumption that the damping is sufficiently strong.
To this purpose, we prove a comparison principle for the damped
wave equation and construct suitable sub- and super-solutions.
This is joint work with Mitsunori Nata.
2013年06月06日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
Chang-Shou Lin 氏 (National Taiwan University)
The Geometry of Critical Points of Green functions On Tori (ENGLISH)
Chang-Shou Lin 氏 (National Taiwan University)
The Geometry of Critical Points of Green functions On Tori (ENGLISH)
[ 講演概要 ]
The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.
We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.
We can generalize our results to a sum of two Green functions.
The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.
We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.
We can generalize our results to a sum of two Green functions.
2012年09月20日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
Bernold Fiedler 氏 (Free University of Berlin)
Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes
(ENGLISH)
Bernold Fiedler 氏 (Free University of Berlin)
Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes
(ENGLISH)
[ 講演概要 ]
Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.
We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.
This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.
Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.
We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.
This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.
2012年01月19日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
Philippe G. LeFloch 氏 (Univ. Paris 6 / CNRS)
Undercompressible shocks and moving phase boundaries
(ENGLISH)
Philippe G. LeFloch 氏 (Univ. Paris 6 / CNRS)
Undercompressible shocks and moving phase boundaries
(ENGLISH)
[ 講演概要 ]
I will present a study of traveling wave solutions to third-order, diffusive-dispersive equations, which arise in the modeling of complex fluid flows and represent regularization-sensitive wave patterns, especially undercompressive shock waves and moving phase boundaries. The qualitative properties of these (possibly oscillatory) traveling waves are well-understood in terms of the so-called kinetic relation, and this has led to a new theory of (nonclassical) solutions to nonlinear hyperbolic systems. Relevant papers are available at the link: www.philippelefloch.org.
I will present a study of traveling wave solutions to third-order, diffusive-dispersive equations, which arise in the modeling of complex fluid flows and represent regularization-sensitive wave patterns, especially undercompressive shock waves and moving phase boundaries. The qualitative properties of these (possibly oscillatory) traveling waves are well-understood in terms of the so-called kinetic relation, and this has led to a new theory of (nonclassical) solutions to nonlinear hyperbolic systems. Relevant papers are available at the link: www.philippelefloch.org.
2011年11月10日(木)
15:00-16:00 数理科学研究科棟(駒場) 128号室
今回はダブルヘッダーのため,開始時間が通常と異なりますのでご注意ください.
森洋一朗 氏 (ミネソタ大学)
電解質および浸透圧調節の細胞生理学とその数理モデル (JAPANESE)
今回はダブルヘッダーのため,開始時間が通常と異なりますのでご注意ください.
森洋一朗 氏 (ミネソタ大学)
電解質および浸透圧調節の細胞生理学とその数理モデル (JAPANESE)
[ 講演概要 ]
細胞体積とイオン濃度の調節は細胞生理学、特に上皮細胞の生理学
において中心的な課題である。細胞体積調節を記述する標準的な数理
モデルはpump-leak model と呼ばれ、数学的には常微分方程式に
代数的な拘束条件の加わった系である。60年代から現在に至るまで
多くの研究者が用いてきたにもかかわらず、その数理的な性質は全く
知られていなかった。本講演では、pump-leak model には熱力学的な
構造があることを解説し、これを利用することで最近得られた解析的な
結果を紹介する。さらにpump-leak model を拡張して得られる偏微分
方程式系とその熱力学的構造について解説する。
細胞体積とイオン濃度の調節は細胞生理学、特に上皮細胞の生理学
において中心的な課題である。細胞体積調節を記述する標準的な数理
モデルはpump-leak model と呼ばれ、数学的には常微分方程式に
代数的な拘束条件の加わった系である。60年代から現在に至るまで
多くの研究者が用いてきたにもかかわらず、その数理的な性質は全く
知られていなかった。本講演では、pump-leak model には熱力学的な
構造があることを解説し、これを利用することで最近得られた解析的な
結果を紹介する。さらにpump-leak model を拡張して得られる偏微分
方程式系とその熱力学的構造について解説する。
2011年11月10日(木)
16:30-17:30 数理科学研究科棟(駒場) 128号室
この日はダブルヘッダーです。
Bernold Fiedler 氏 (Free University of Berlin)
Schoenflies spheres in Sturm attractors (ENGLISH)
この日はダブルヘッダーです。
Bernold Fiedler 氏 (Free University of Berlin)
Schoenflies spheres in Sturm attractors (ENGLISH)
[ 講演概要 ]
In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.
For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.
In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.
For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.
2011年06月30日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
鹿島 洋平 氏 (東京大学大学院数理科学研究科)
3次元空間における第2種超電導の巨視的モデルについて (JAPANESE)
鹿島 洋平 氏 (東京大学大学院数理科学研究科)
3次元空間における第2種超電導の巨視的モデルについて (JAPANESE)
[ 講演概要 ]
3次元空間に置かれた第2種超電導体のまわりの電磁場の巨視的なふるまいを記述するモデルについて考察する.超電導を特徴付ける非線型の電場と電流密度の関係式をマックスウェル方程式系と組み合わせることで磁場に関する時間発展の方程式を導くことができる.この非線型のオームの法則としてビーンモデルが応用上典型的に用いられるが,超電導体の内部でピン止めされた磁束が動き出すとき一般の3次元の問題では電流と平行でない電場が現れることが予測され,ビーンモデルの仮定に反してしまう.3次元のより現実的な巨視的モデルの候補としては,Double Critical-state Modelが1980年代に提案されている.本講演ではこれらの巨視的モデルを3次元の問題に採用して導いた発展方程式の解の存在と有限要素法による離散化の方法を解説し,あわせて数値計算例を提示する.
3次元空間に置かれた第2種超電導体のまわりの電磁場の巨視的なふるまいを記述するモデルについて考察する.超電導を特徴付ける非線型の電場と電流密度の関係式をマックスウェル方程式系と組み合わせることで磁場に関する時間発展の方程式を導くことができる.この非線型のオームの法則としてビーンモデルが応用上典型的に用いられるが,超電導体の内部でピン止めされた磁束が動き出すとき一般の3次元の問題では電流と平行でない電場が現れることが予測され,ビーンモデルの仮定に反してしまう.3次元のより現実的な巨視的モデルの候補としては,Double Critical-state Modelが1980年代に提案されている.本講演ではこれらの巨視的モデルを3次元の問題に採用して導いた発展方程式の解の存在と有限要素法による離散化の方法を解説し,あわせて数値計算例を提示する.
2011年06月09日(木)
16:30-18:00 数理科学研究科棟(駒場) 128号室
時間が普段と異なりますのでご注意ください
望月清 氏 (東京都立大学 名誉教授)
Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)
時間が普段と異なりますのでご注意ください
望月清 氏 (東京都立大学 名誉教授)
Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)
[ 講演概要 ]
We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.
We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.
2011年05月26日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
深尾武史 氏 (京都教育大学)
Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)
深尾武史 氏 (京都教育大学)
Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)
[ 講演概要 ]
In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,
depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.
In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,
depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.
2011年05月19日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
伊藤真吾 氏 (東京理科大学)
波束変換を用いて定義される波面集合とその応用 (JAPANESE)
伊藤真吾 氏 (東京理科大学)
波束変換を用いて定義される波面集合とその応用 (JAPANESE)
[ 講演概要 ]
本講演ではC'ordoba-Feffermanによって導入された波束変換(wave packet transform)を用いて、ある波面集合を定義し、それを特異性伝播の問題に応用する。特異性の伝播とは、双曲型方程式の特徴の一つで方程式の解の初期値が持つ特異性が、時間の経過とともにある曲線に沿って伝わっていく現象のことである。なお、本研究は東京理科大学の加藤圭一氏ならびに小林政晴氏との共同研究である。
本講演ではC'ordoba-Feffermanによって導入された波束変換(wave packet transform)を用いて、ある波面集合を定義し、それを特異性伝播の問題に応用する。特異性の伝播とは、双曲型方程式の特徴の一つで方程式の解の初期値が持つ特異性が、時間の経過とともにある曲線に沿って伝わっていく現象のことである。なお、本研究は東京理科大学の加藤圭一氏ならびに小林政晴氏との共同研究である。
2011年04月14日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
Marek FILA 氏 (Comenius University (Slovakia))
Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)
Marek FILA 氏 (Comenius University (Slovakia))
Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)
[ 講演概要 ]
We study the existence of connecting orbits for the Fujita equation
u_t=\\Delta u+u^p
with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.
We study the existence of connecting orbits for the Fujita equation
u_t=\\Delta u+u^p
with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.
2011年02月24日(木)
16:00-18:10 数理科学研究科棟(駒場) 002号室
Arnaud Ducrot 氏 (University of Bordeaux 2) 16:00-17:00
Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)
Some open problems in PDE control (ENGLISH)
Arnaud Ducrot 氏 (University of Bordeaux 2) 16:00-17:00
Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)
[ 講演概要 ]
This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological viewpoint, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established.
Enrique Zuazua 氏 (Basque Center for Applied Mathematics) 17:10-18:10This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological viewpoint, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established.
Some open problems in PDE control (ENGLISH)
[ 講演概要 ]
The field of PDE control has experienced a great progress in the last decades, developing new theories and tools that have also influenced other disciplines as Inverse Problem and Optimal Design Theories and Numerical Analysis. PDE control arises in most applications ranging from classical problems in fluid mechanics or structural engineering to modern molecular design experiments.
From a mathematical viewpoint the problems arising in this field are extremely challenging since the existing theory of existence and uniqueness of solutions and the corresponding numerical schemes is insufficient when addressing realistic control problems. Indeed, an efficient controller requires of an in depth understanding of how solutions depend on the various parameters of the problem (shape of the domain, time of control, coefficients of the equation, location
of the controller, nonlinearity in the equation,...)
In this lecture we shall briefly discuss some important advances and some challenging open problems. All of them shear some features. In particular they are simple to state and very likely hard to solve. We shall discuss in particular:
1.- Semilinear wave equations and their control properties.
2.- Microlocal optimal design of wave processes
3.- Sharp observability estimates for heat processes.
4.- Robustness on the control of finite-dimensional systems.
5.- Unique continuation for discrete elliptic models
6.- Control of Kolmogorov equations and other hypoelliptic models.
The field of PDE control has experienced a great progress in the last decades, developing new theories and tools that have also influenced other disciplines as Inverse Problem and Optimal Design Theories and Numerical Analysis. PDE control arises in most applications ranging from classical problems in fluid mechanics or structural engineering to modern molecular design experiments.
From a mathematical viewpoint the problems arising in this field are extremely challenging since the existing theory of existence and uniqueness of solutions and the corresponding numerical schemes is insufficient when addressing realistic control problems. Indeed, an efficient controller requires of an in depth understanding of how solutions depend on the various parameters of the problem (shape of the domain, time of control, coefficients of the equation, location
of the controller, nonlinearity in the equation,...)
In this lecture we shall briefly discuss some important advances and some challenging open problems. All of them shear some features. In particular they are simple to state and very likely hard to solve. We shall discuss in particular:
1.- Semilinear wave equations and their control properties.
2.- Microlocal optimal design of wave processes
3.- Sharp observability estimates for heat processes.
4.- Robustness on the control of finite-dimensional systems.
5.- Unique continuation for discrete elliptic models
6.- Control of Kolmogorov equations and other hypoelliptic models.
2011年02月17日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Thomas Giletti 氏 (University of Paul Cezanne (Marseilles))
Study of propagation phenomena in some reaction-diffusion systems (ENGLISH)
Thomas Giletti 氏 (University of Paul Cezanne (Marseilles))
Study of propagation phenomena in some reaction-diffusion systems (ENGLISH)
[ 講演概要 ]
This talk deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction-diffusion system with losses inside the domain, which has numerous applications in various fields ranging from chemical and biological contexts to combusion. Under some KPP type hypotheses, the existence of a continuum of admissible speeds for traveling waves can be shown, thus generalizing the single equation case. Lastly, by considering losses concentrated near the edge of the domain, those results can be compared with those of the boundary losses case.
This talk deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction-diffusion system with losses inside the domain, which has numerous applications in various fields ranging from chemical and biological contexts to combusion. Under some KPP type hypotheses, the existence of a continuum of admissible speeds for traveling waves can be shown, thus generalizing the single equation case. Lastly, by considering losses concentrated near the edge of the domain, those results can be compared with those of the boundary losses case.
2011年02月10日(木)
15:30-16:30 数理科学研究科棟(駒場) 002号室
時間が通常と異なりますのでご注意ください.
Jean-Michel Coron 氏 (University of Paris 6)
Control and nonlinearity (ENGLISH)
時間が通常と異なりますのでご注意ください.
Jean-Michel Coron 氏 (University of Paris 6)
Control and nonlinearity (ENGLISH)
[ 講演概要 ]
We present methods to study the controllability and the stabilizability of nonlinear control systems. The emphasis is put on specific phenomena due to the nonlinearities. In particular we study cases where the nonlinearities are essential for the controllability or the stabilizability.
We illustrate these methods on control systems modeled by ordinary differential equations or partial differential equations (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg de Vries equations).
We present methods to study the controllability and the stabilizability of nonlinear control systems. The emphasis is put on specific phenomena due to the nonlinearities. In particular we study cases where the nonlinearities are essential for the controllability or the stabilizability.
We illustrate these methods on control systems modeled by ordinary differential equations or partial differential equations (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg de Vries equations).
2011年01月27日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Nitsan Ben-Gal 氏 (The Weizmann Institute of Science)
Attraction at infinity: Constructing non-compact global attractors in the slowly non-dissipative realm (ENGLISH)
Nitsan Ben-Gal 氏 (The Weizmann Institute of Science)
Attraction at infinity: Constructing non-compact global attractors in the slowly non-dissipative realm (ENGLISH)
[ 講演概要 ]
One of the primary tools for understanding the much-studied realm of reaction-diffusion equations is the global attractor, which provides us with a qualitative understanding of the governing behaviors of solutions to the equation in question. Nevertheless, the classic global attractor for such systems is defined to be compact, and thus attractor theory has previously excluded such analysis from being applied to non-dissipative reaction-diffusion equations.
In this talk I will present recent results in which I developed a non-compact analogue to the classical global attractor, and will discuss the methods derived in order to obtain a full decomposition of the non-compact global attractor for a slowly non-dissipative reaction-diffusion equation. In particular, attention will be paid to the nodal property techniques and reduction methods which form a critical underpinning of asymptotics research in both dissipative and non-dissipative evolutionary equations. I will discuss the concepts of the ‘completed inertial manifold’ and ‘non-compact global attractor’, and show how these in particular allow us to produce equivalent results for a class of slowly non-dissipative equations as have been achieved for dissipative equations. Additionally, I will address the behavior of solutions to slowly non-dissipative equations approaching and at infinity, the realm which presents both the challenges and rewards of removing the necessity of dissipativity.
One of the primary tools for understanding the much-studied realm of reaction-diffusion equations is the global attractor, which provides us with a qualitative understanding of the governing behaviors of solutions to the equation in question. Nevertheless, the classic global attractor for such systems is defined to be compact, and thus attractor theory has previously excluded such analysis from being applied to non-dissipative reaction-diffusion equations.
In this talk I will present recent results in which I developed a non-compact analogue to the classical global attractor, and will discuss the methods derived in order to obtain a full decomposition of the non-compact global attractor for a slowly non-dissipative reaction-diffusion equation. In particular, attention will be paid to the nodal property techniques and reduction methods which form a critical underpinning of asymptotics research in both dissipative and non-dissipative evolutionary equations. I will discuss the concepts of the ‘completed inertial manifold’ and ‘non-compact global attractor’, and show how these in particular allow us to produce equivalent results for a class of slowly non-dissipative equations as have been achieved for dissipative equations. Additionally, I will address the behavior of solutions to slowly non-dissipative equations approaching and at infinity, the realm which presents both the challenges and rewards of removing the necessity of dissipativity.
2010年07月08日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Anna Vainchtein 氏 (University of Pittsburgh, Department of Mathematics)
Effect of nonlinearity on the steady motion of a twinning dislocation (ENGLISH)
Anna Vainchtein 氏 (University of Pittsburgh, Department of Mathematics)
Effect of nonlinearity on the steady motion of a twinning dislocation (ENGLISH)
[ 講演概要 ]
We consider the steady motion of a twinning dislocation in a Frenkel-Kontorova lattice with a double-well substrate potential that has a non-degenerate spinodal region. Semi-analytical traveling wave solutions are constructed for the piecewise quadratic potential, and their stability and further effects of nonlinearity are investigated numerically. We show that the width of the spinodal region and the nonlinearity of the potential have a significant effect on the dislocation kinetics, resulting in stable steady motion in some low-velocity intervals and lower propagation stress. We also conjecture that a stable steady propagation must correspond to an increasing portion of the kinetic relation between the applied stress and dislocation velocity.
We consider the steady motion of a twinning dislocation in a Frenkel-Kontorova lattice with a double-well substrate potential that has a non-degenerate spinodal region. Semi-analytical traveling wave solutions are constructed for the piecewise quadratic potential, and their stability and further effects of nonlinearity are investigated numerically. We show that the width of the spinodal region and the nonlinearity of the potential have a significant effect on the dislocation kinetics, resulting in stable steady motion in some low-velocity intervals and lower propagation stress. We also conjecture that a stable steady propagation must correspond to an increasing portion of the kinetic relation between the applied stress and dislocation velocity.
2010年06月24日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
GCOE 共催
村川 秀樹 氏 (富山大学大学院理工学研究部)
非線形拡散問題の反応拡散系近似 (JAPANESE)
GCOE 共催
村川 秀樹 氏 (富山大学大学院理工学研究部)
非線形拡散問題の反応拡散系近似 (JAPANESE)
[ 講演概要 ]
氷の融解・水の凝固の過程を記述するステファン問題、地下水の流れを表す多孔質媒体流方程式、2種生物種の競合問題における互いの動的な干渉作用を記述する重定-川崎-寺本交差拡散系など、様々な問題を含む非線形拡散問題を取り扱う。本講演では、非線形拡散問題の解が、拡散が線形である半線形反応拡散系の解により近似されることを示す。この結果は、非線形拡散問題の解構造が、ある種の半線形反応拡散系の中に再現されることを示唆するものである。一般に、非線形問題を扱うよりも半線形問題を取り扱う方が容易であるため、本研究は非線形問題の解析や数値解析に応用できることが期待される。
氷の融解・水の凝固の過程を記述するステファン問題、地下水の流れを表す多孔質媒体流方程式、2種生物種の競合問題における互いの動的な干渉作用を記述する重定-川崎-寺本交差拡散系など、様々な問題を含む非線形拡散問題を取り扱う。本講演では、非線形拡散問題の解が、拡散が線形である半線形反応拡散系の解により近似されることを示す。この結果は、非線形拡散問題の解構造が、ある種の半線形反応拡散系の中に再現されることを示唆するものである。一般に、非線形問題を扱うよりも半線形問題を取り扱う方が容易であるため、本研究は非線形問題の解析や数値解析に応用できることが期待される。
2010年06月10日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Christian Klingenberg 氏 (Wuerzburg 大学 )
Hydrodynamic limit of microscopic particle systems to conservation laws to fluid models
Christian Klingenberg 氏 (Wuerzburg 大学 )
Hydrodynamic limit of microscopic particle systems to conservation laws to fluid models
[ 講演概要 ]
In this talk we discuss the hydrodynamic limit of a microscopic description of a fluid to its macroscopic PDE description.
In the first part we consider flow through porous media, i.e. the macroscopic description is a scalar conservation law. Here the new feature is that we allow sudden changes in porosity and thereby the flux may have discontinuities in space. Microscopically this is described through an interacting particle system having only one conserved quantity, namely the total mass. Macroscopically this gives rise to a scalar conservation laws with space dependent flux functions
u_t + f(u, x)_x = 0 .
We are able to derive the PDE together with an entropy condition as a hydrodynamic limit from a microscopic interacting particle system.
In the second part we consider a Hamiltonian system with boundary conditions. Microscopically this is described through a system of coupled oscillators. Macroscopically this will lead to a system of conservation laws, namely the p-system. The proof of the hydrodynamic limit is restricted to smooth solutions. The new feature is that we can derive this with boundary conditions.
In this talk we discuss the hydrodynamic limit of a microscopic description of a fluid to its macroscopic PDE description.
In the first part we consider flow through porous media, i.e. the macroscopic description is a scalar conservation law. Here the new feature is that we allow sudden changes in porosity and thereby the flux may have discontinuities in space. Microscopically this is described through an interacting particle system having only one conserved quantity, namely the total mass. Macroscopically this gives rise to a scalar conservation laws with space dependent flux functions
u_t + f(u, x)_x = 0 .
We are able to derive the PDE together with an entropy condition as a hydrodynamic limit from a microscopic interacting particle system.
In the second part we consider a Hamiltonian system with boundary conditions. Microscopically this is described through a system of coupled oscillators. Macroscopically this will lead to a system of conservation laws, namely the p-system. The proof of the hydrodynamic limit is restricted to smooth solutions. The new feature is that we can derive this with boundary conditions.
2010年04月22日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Jens Starke 氏 (デンマーク工科大学)
Deterministic and stochastic modelling of catalytic surface processes (ENGLISH)
Jens Starke 氏 (デンマーク工科大学)
Deterministic and stochastic modelling of catalytic surface processes (ENGLISH)
[ 講演概要 ]
Three levels of modelling, the microscopic, the mesoscopic and the macroscopic level are discussed for the CO oxidation on low-index platinum single crystal surfaces. The introduced models on the microscopic and mesoscopic level are stochastic while the model on the macroscopic level is deterministic. The macroscopic description can be derived rigorously for low pressure conditions as limit of the stochastic many particle model for large particle numbers. This is in correspondence with the successful description of experiments under low pressure conditions by deterministic reaction-diffusion equations while for intermediate pressures phenomena of stochastic origin can be observed in experiments. The introduced models include a new approach for the platinum phase transition which allows for a unification of existing models for Pt(100) and Pt(110).
The rich nonlinear dynamical behaviour of the macroscopic reaction kinetics is investigated and shows good agreement with low pressure experiments. Furthermore, for intermediate pressures, noise-induced pattern formation, so-called raindrop patterns which are not captured by earlier models, can be reproduced and are shown in simulations.
This is joint work with M. Eiswirth, H. Rotermund, G. Ertl,
Frith Haber Institut, Berlin, K. Oelschlaeger, University of
Heidelberg and C. Reichert, INSA, Lyon.
Three levels of modelling, the microscopic, the mesoscopic and the macroscopic level are discussed for the CO oxidation on low-index platinum single crystal surfaces. The introduced models on the microscopic and mesoscopic level are stochastic while the model on the macroscopic level is deterministic. The macroscopic description can be derived rigorously for low pressure conditions as limit of the stochastic many particle model for large particle numbers. This is in correspondence with the successful description of experiments under low pressure conditions by deterministic reaction-diffusion equations while for intermediate pressures phenomena of stochastic origin can be observed in experiments. The introduced models include a new approach for the platinum phase transition which allows for a unification of existing models for Pt(100) and Pt(110).
The rich nonlinear dynamical behaviour of the macroscopic reaction kinetics is investigated and shows good agreement with low pressure experiments. Furthermore, for intermediate pressures, noise-induced pattern formation, so-called raindrop patterns which are not captured by earlier models, can be reproduced and are shown in simulations.
This is joint work with M. Eiswirth, H. Rotermund, G. Ertl,
Frith Haber Institut, Berlin, K. Oelschlaeger, University of
Heidelberg and C. Reichert, INSA, Lyon.
2010年04月15日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Alberto Tesei 氏 (University of Rome 1)
Long-time behaviour of solutions of a forward-backward parabolic equation
Alberto Tesei 氏 (University of Rome 1)
Long-time behaviour of solutions of a forward-backward parabolic equation
[ 講演概要 ]
We discuss some recent results concerning the asymptotic behaviour of entropy measure-valued solutions for a class of ill-posed forward-backward parabolic equations, which arise in the theory of phase transitions.
We discuss some recent results concerning the asymptotic behaviour of entropy measure-valued solutions for a class of ill-posed forward-backward parabolic equations, which arise in the theory of phase transitions.
2010年02月18日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
Bendong LOU 氏 (同済大学)
Homogenization limit of a parabolic equation with nonlinear boundary conditions
Bendong LOU 氏 (同済大学)
Homogenization limit of a parabolic equation with nonlinear boundary conditions
[ 講演概要 ]
We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:
"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".
We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:
"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".
2010年01月28日(木)
16:00-17:30 数理科学研究科棟(駒場) 002号室
清水扇丈 氏 (静岡大学理学部)
相転移を伴う非圧縮性2相流の線形化問題について
清水扇丈 氏 (静岡大学理学部)
相転移を伴う非圧縮性2相流の線形化問題について
[ 講演概要 ]
氷が常圧で0度以上になると水になるなどの相転移を伴う非圧縮性2相流に対し,質量保存則, 運動量保存則, エネルギー保存則を界面を含む系全体に適用し, 線形化した方程式系について考察する. 本講演では, 線形化方程式系のL_p-L_q 最大正則性定理について述べる.
密度が異なる場合は, 法線方向の高さ関数は表面張力つき2相Stokes問題の高さ関数と同じ正則性をもち, 系は流速が支配するのに対し,密度が等しい場合は, Gibbs-Thomson補正された表面張力つき2相Stefan問題の高さ関数と同じ正則性をもち, 系は温度が支配する.
氷が常圧で0度以上になると水になるなどの相転移を伴う非圧縮性2相流に対し,質量保存則, 運動量保存則, エネルギー保存則を界面を含む系全体に適用し, 線形化した方程式系について考察する. 本講演では, 線形化方程式系のL_p-L_q 最大正則性定理について述べる.
密度が異なる場合は, 法線方向の高さ関数は表面張力つき2相Stokes問題の高さ関数と同じ正則性をもち, 系は流速が支配するのに対し,密度が等しい場合は, Gibbs-Thomson補正された表面張力つき2相Stefan問題の高さ関数と同じ正則性をもち, 系は温度が支配する.
2010年01月21日(木)
16:00-17:30 数理科学研究科棟(駒場) 122号室
Danielle Hilhorst 氏 (パリ南大学 / CNRS)
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
Danielle Hilhorst 氏 (パリ南大学 / CNRS)
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
[ 講演概要 ]
We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.
We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.
